Toward the $p$-adic Hodge parameters in the potentially crystalline representations of $\mathrm{GL}_n$

Abstract: Let $p$ be a prime number, $n$ an integer $\geq 2$, and $L$ a finite extension of $\mathrm{Q}p$. Let $ρ_L$ be an $n$-dimensional (non-critical but not necessary generic) potentially crystalline $p$-adic Galois representation of the absolute Galois groups of $L$ of regular Hodge-Tate weights. By generalizing the previous results and strategy for the crystabelline case of Ding and the recent work of Breuil-Ding, we construct an explicit locally analytic representation $π{1}(ρL)$, and describe explicitly the information of Hodge filtration of $ρ_L$ it determines. When $ρ_L$ comes from a patched $p$-adic automorphic representation, we show that $π{1}(ρ_L)$ is a subrepresentation of the $\mathrm{GL}_n(L)$-representation globally associated to $ρ_L$, under some mild hypothesis.